SUPERINTEGERS AND HYPERNUMBERS

SUPERINTEGERS AND HYPERNUMBERS

Now I will describe the concept of a superinteger. A superinteger is a neomathematical entity of sufficient weight, complexity, and structural coherence that, it is in itself invariant under various transformations, and it can, under certain transformations, generate other neomathematical entities.

Except when we introduce a special symbol, we will specify a superinteger by a polygonal symbol,such as I orK

I for example, is the superinteger for identity, and we have such transformations as

where the superinteger is shown as generating the initial self relation of a subpoint.

(The transformations are shown vertically because of the limitations of html; it would ordinarily seem more natural to write them horizontally.)

The neomath surfaces well in the duality between

e I

and

where e is coevalence.

It is clear that e is an instance of a superinteger also, e.

We can also see that our generalized transformation,

, is also an instance of a superinteger

We can form the superinteger for twists, T, where for example,

We can form a superinteger for triadic dualism

The superinteger denotes the concept with it names, as well as its possible symbolic embodiments, and their implicit concatenations. If we wish to ascend to the level of physical or psychophysical embodiment of a concept, then I believe it is valid to introduce a new concept, called 'hypernumber'. Essentially, a superinteger exists in the plane of the paper or on the screen where it appears, while a hypernumber may also exist in the extended real world or as a Platonic form. A hypernumber may intersect the plane of the paper as a superinteger. However, the superinteger is a rather special neomathematical object, whereas the hypernumber is also a nonmathematical generalization. For example, it might be that the intersection of a hypernumber with the plane of the paper would include a superinteger and a complex diagram, with perhaps also prose describing the hypernumber. And since hypernumbers are also projected to have an existence independent of their realization in thought, perhaps, then we expect hypernumbers for which their is no superinteger. Each superinteger has a hypernumber.

We can consider a hypernumber defined by the following statement: 'This statement represents a logical entity which defined itself by existing.' Name this hypernumber D. The ontology of D lies in logical space, thus it also has some presence in psychophysical space, but may either be very thin or very thick in physical space. D is the hypernumber of I.

Now we will begin construction of another hypernumber. We are trying in this instance to formulate the concept of otherness. That is, given any formulation of any entity or class of entities whatever, we mean something else. There is a linguistic discontinuity here, but no paradox. Think of a concept formulation which might be considered as a bounded region in concept space; we mean somewhere else, outside of that bounded region, although if we could mention a particular place, that is no longer it. We are helped if we cannot formulate concept space. Let us call this hypernumber strangeness and denote it by its superinteger, S.

A great many mathematical entities can in some way be approximated by linear structures or by collections of linear structures. Call this hypernumber 'star linear' and denote it by its superinteger L. One aspect of star linearity is to be approachable, perhaps to be reached or approximated. If we go to the very opposite situation, where something is unreachable, unapproachable, we can make this a hypernumber also,and give it the superinteger

Sometimes things happen which appear to have meaning. These may be highly structured core events which control much of the surrounding reality. Good examples include the elementary catastrophes or Rene Thom. Suppose that additionally we add in a concept of value, good, benefit or just plain luck. We make of this a new hypernumber and denote it by its superinteger

Another hypernumber I would like to introduce concerns the possibility of events. An event may either occur or not occur (or both in the Everett interpretation). We call the possibility of anything happening, the hypernumber and the superinteger thereof, 'aleatory 1/2', which we symbolize by 1/2. Every event has possibility of occurrence 1/2, whatever its probability of occurrence.

Another hypernumber one might find useful is the concept, 'hypercenter', denoted by C. A hypercenter is defined to be an entity with a bounded exterior but unbounded interior.

A point has bounded exterior but an unbounded interior structure of subpoint relations.

Another example of hypercenter is the individual consciousness.

If we consider experienced reality, it appears that much of what happens is causally connected to earlier and later events. Often this causality can be described by smooth functions, by solutions to boundary value problems, or in terms of the elementary catastrophes. Particular events in this reality do not, in general, seem to have an entirely independent existence. Now suppose there is a hypernumber which can affect change, be causation, but is itself entirely un caused, not subject to external causation. Existence would be one facet of this hypernumber.

We can imagine changes in this hypernumber, as long as they are not caused changes.

Let us call this hypernumber k-space and denote it by its superinteger K. We might use k for a particular occurrence of K.

One aspect of K-space is to be arbitrary. While un caused, it is conceivable that K-space is co generated with human will.

Now we can write and describe a variety of superinteger, and implicitly hypernumber, transformations

In the diagram above, K gives rise to identity.

Luck comes from the inapproachable limit of K-space

K-space transmits existence.

What sources K

S 1/2

Anything can exist, as well as not.

1/2

Even good fortune.

Joe Staley
back to Neomathematics

back to Mind Alive